143 research outputs found
Axiomatics for the external numbers of nonstandard analysis
Neutrices are additive subgroups of a nonstandard model of the real numbers.
An external number is the algebraic sum of a nonstandard real number and a
neutrix. Due to the stability by some shifts, external numbers may be seen as
mathematical models for orders of magnitude. The algebraic properties of
external numbers gave rise to the so-called solids, which are extensions of
ordered fields, having a restricted distributivity law. However, necessary and
sufficient conditions can be given for distributivity to hold. In this article
we develop an axiomatics for the external numbers. The axioms are similar to,
but mostly somewhat weaker than the axioms for the real numbers and deal with
algebraic rules, Dedekind completeness and the Archimedean property. A
structure satisfying these axioms is called a complete arithmetical solid. We
show that the external numbers form a complete arithmetical solid, implying the
consistency of the axioms presented. We also show that the set of precise
elements (elements with minimal magnitude) has a built-in nonstandard model of
the rationals. Indeed the set of precise elements is situated between the
nonstandard rationals and the nonstandard reals whereas the set of non-precise
numbers is completely determined
Realizability with Stateful Computations for Nonstandard Analysis
In this paper we propose a new approach to realizability interpretations for nonstandard arithmetic. We deal with nonstandard analysis in the context of intuitionistic realizability, focusing on the Lightstone-Robinson construction of a model for nonstandard analysis through an ultrapower. In particular, we consider an extension of the ?-calculus with a memory cell, that contains an integer (the state), in order to indicate in which slice of the ultrapower ?^{?} the computation is being done. We shall pay attention to the nonstandard principles (and their computational content) obtainable in this setting. We then discuss how this product could be quotiented to mimic the Lightstone-Robinson construction
Stateful Realizers for Nonstandard Analysis
In this paper we propose a new approach to realizability interpretations for
nonstandard arithmetic. We deal with nonstandard analysis in the context of
(semi)intuitionistic realizability, focusing on the Lightstone-Robinson
construction of a model for nonstandard analysis through an ultrapower. In
particular, we consider an extension of the -calculus with a memory
cell, that contains an integer (the state), in order to indicate in which slice
of the ultrapower the computation is being done. We pay
attention to the nonstandard principles (and their computational content)
obtainable in this setting. In particular, we give non-trivial realizers to
Idealization and a non-standard version of the LLPO principle. We then discuss
how to quotient this product to mimic the Lightstone-Robinson construction
Flexible involutive meadows
We investigate a notion of inverse for neutrices inspired by Van den Berg and
Koudjeti's decomposition of a neutrix as the product of a real number and an
idempotent neutrix. We end up with an algebraic structure that can be
characterized axiomatically and generalizes involutive meadows. The latter are
algebraic structures where the inverse for multiplication is a total operation.
As it turns out, the structures satisfying the axioms of flexible involutive
meadows are of interest beyond nonstandard analysis
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